{"paper":{"title":"On the number of types in sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Micha{\\l} Pilipczuk, Sebastian Siebertz, Szymon Toru\\'nczyk","submitted_at":"2017-05-25T19:22:25Z","abstract_excerpt":"We prove that for every class of graphs $\\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\\phi(\\bar x,\\bar y)$, whenever one draws a graph $G\\in \\mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\\bar y|}$ which are of the form $\\{\\bar v\\in A^{|\\bar y|}\\, \\colon\\, G\\models\\phi(\\bar u,\\bar v)\\}$ for some valuation $\\bar u$ of $\\bar x$ in $G$ is bounded by $\\mathcal{O}(|A|^{|\\bar x|+\\epsilon})$, for every $\\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09336","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}